# Holonomic basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {e1, ..., en} defined at every point P of a region of the manifold as

${\displaystyle \mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac {\delta \mathbf {s} }{\delta x^{\alpha }}},}$

where δs is the infinitesimal displacement vector between the point P and a nearby point Q whose coordinate separation from P is δxα along the coordinate curve xα (i.e. the curve on the manifold through P for which the local coordinate xα varies and all other coordinates are constant).[1]

It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve C on the manifold defined by xα(λ) with the tangent vector u = uαeα, where uα = dxα/, and a function f(xα) defined in a neighbourhood of C, the variation of f along C can be written as

${\displaystyle {\frac {df}{d\lambda }}={\frac {dx^{\alpha }}{d\lambda }}{\frac {\partial f}{\partial x^{\alpha }}}=u^{\alpha }{\frac {\partial f}{\partial x^{\alpha }}}.}$

Since we have that u = uαeα, the identification is often made between a coordinate basis vector eα and the partial derivative operator /xα, under the interpretation of all vector relations as equalities between operators acting on scalar quantities.[2]

A local condition for a basis {e1, ..., en} to be holonomic is that all mutual Lie derivatives vanish:[3]

${\displaystyle \left[\mathbf {e} _{\alpha },\mathbf {e} _{\beta }\right]={\mathcal {L}}_{\mathbf {e} _{\alpha }}\mathbf {e} _{\beta }=0.}$

A basis that is not holonomic is called a non-holonomic or non-coordinate basis.

Given a metric tensor g on a manifold M, it is in general not possible to find a coordinate basis that is orthonormal in any open region U of M.[4] An obvious exception is when M is the real coordinate space Rn considered as a manifold with g being the Euclidean metric δijeiej at every point.

## References

1. ^ M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006), General Relativity: An Introduction for Physicists, Cambridge University Press, p. 57
2. ^ T. Padmanabhan (2010), Gravitation: Foundations and Frontiers, Cambridge University Press, p. 25
3. ^ Roger Penrose; Wolfgang Rindler, Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, pp. 197–199
4. ^ Bernard F. Schutz (1980), Geometrical Methods of Mathematical Physics, Cambridge University Press, p. 69, ISBN 9780521298872